Class 6 Mathematics Notes Chapter 13 (Symmetry) – Mathematics Book

Detailed Notes with MCQs of Chapter 13, 'Symmetry', from your NCERT Class 6 Mathematics textbook. This is a visually interesting chapter and the concepts are fundamental, often appearing in various competitive exams in logical reasoning or spatial ability sections. Pay close attention as we break it down.

Chapter 13: Symmetry - Detailed Notes for Exam Preparation

1. Introduction to Symmetry

• What is Symmetry? Symmetry refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry means that one shape becomes exactly like another when you move it in some way: turn, flip or slide. For this chapter (Class 6 level), we primarily focus on line symmetry.
• Informal Idea: If you can fold a picture or shape in half so that the two halves match exactly, then the shape is symmetrical, and the fold line is the line of symmetry.
• Real-life Examples: Butterflies, leaves, human faces (approximately), architectural designs (like the Taj Mahal), letters of the alphabet, geometric shapes, etc.

2. Line Symmetry (Reflectional Symmetry)

• Definition: A figure has line symmetry if a line can be drawn dividing the figure into two identical parts (mirror images of each other).
• Line of Symmetry (Axis of Symmetry): This is the imaginary line along which you can fold the figure so that both halves coincide exactly.
  • Think of it as a mirror line. If you place a mirror along the line of symmetry, the reflection of one half would perfectly match the other half.
• Properties:
  • The line of symmetry divides the shape into two congruent (identical in shape and size) halves.
  • Each point on one side of the line has a corresponding point on the other side, equidistant from the line.

3. Identifying Lines of Symmetry in Shapes

• Shapes with One Line of Symmetry:

  • Isosceles Triangle (line joining the vertex between equal sides to the midpoint of the opposite base)
  • Kite (along the longer diagonal, usually)
  • Semicircle
  • Arrowhead
  • Letters like A, M, T, U, V, W, Y (Vertical Line)
  • Letters like B, C, D, E, K (Horizontal Line)

• Shapes with Two Lines of Symmetry:

  • Rectangle (lines joining midpoints of opposite sides)
  • Rhombus (the two diagonals)
  • Letters like H, I, X (One vertical, one horizontal)

• Shapes with Three Lines of Symmetry:

  • Equilateral Triangle (lines joining each vertex to the midpoint of the opposite side)

• Shapes with Four Lines of Symmetry:

  • Square (two diagonals and two lines joining midpoints of opposite sides)

• Shapes with More Than Two (Multiple) Lines of Symmetry:

  • Regular Polygons: A regular polygon has as many lines of symmetry as it has sides.
    • Equilateral Triangle (Regular 3-gon): 3 lines
    • Square (Regular 4-gon): 4 lines
    • Regular Pentagon (5-gon): 5 lines
    • Regular Hexagon (6-gon): 6 lines
  • Circle: A circle has infinite lines of symmetry. Any line passing through the center of the circle is a line of symmetry.

• Shapes with No Lines of Symmetry:

  • Scalene Triangle (all sides different)
  • Parallelogram (unless it's a rectangle, rhombus, or square)
  • Trapezium (unless it's an isosceles trapezium)
  • Letters like F, G, J, L, N, P, Q, R, S, Z

4. Reflection and Symmetry

• Line symmetry is closely related to reflection. The line of symmetry acts as a mirror.
• One half of the symmetrical figure is the mirror image (reflection) of the other half.
• Key aspects of reflection:
  • The image is the same distance from the mirror line as the object.
  • The size of the image is the same as the size of the object.
  • The image is laterally inverted (like seeing letters reversed in a mirror), but in the context of folding a shape onto itself, this just means the halves match perfectly.

5. Practical Construction/Verification

• Ink-blot Devils: Folding a paper with an ink blot creates symmetrical patterns.
• Paper Folding: Folding paper cutouts is the easiest way to find lines of symmetry for simple shapes.

Key Takeaways for Exams:

• Understand the definition of line symmetry and the line of symmetry.
• Be able to identify the number of lines of symmetry for common geometric shapes (triangles, quadrilaterals, regular polygons, circle).
• Be able to identify lines of symmetry (horizontal, vertical) in capital letters of the English alphabet.
• Recognize shapes that have no lines of symmetry.
• Understand the connection between line symmetry and reflection (mirror image).
• Remember the special case: A circle has infinite lines of symmetry.
• Remember the rule for regular polygons: Number of sides = Number of lines of symmetry.

Multiple Choice Questions (MCQs) on Symmetry

Here are 10 MCQs based on Chapter 13 to test your understanding:

1. A line that divides a figure into two identical halves is called:
(a) A diagonal
(b) A median
(c) A line of symmetry
(d) A perpendicular bisector

2. How many lines of symmetry does a rectangle have?
(a) 1
(b) 2
(c) 4
(d) Infinite

3. Which of the following shapes has exactly one line of symmetry?
(a) Square
(b) Equilateral Triangle
(c) Isosceles Triangle
(d) Circle

4. Which capital letter of the English alphabet has NO line of symmetry?
(a) A
(b) H
(c) O
(d) F

5. A regular hexagon has how many lines of symmetry?
(a) 3
(b) 4
(c) 6
(d) 8

6. Which shape has infinite lines of symmetry?
(a) Square
(b) Circle
(c) Regular Pentagon
(d) Equilateral Triangle

7. The number of lines of symmetry in a scalene triangle is:
(a) 0
(b) 1
(c) 2
(d) 3

8. Line symmetry is also known as:
(a) Rotational Symmetry
(b) Point Symmetry
(c) Translational Symmetry
(d) Reflectional Symmetry

9. Which of the following letters has only a horizontal line of symmetry?
(a) H
(b) X
(c) D
(d) M

10. If you fold a square along one of its diagonals, the two halves:
(a) Do not match
(b) Match exactly
(c) Form a rectangle
(d) Form two smaller squares

Answer Key for MCQs:

1. (c) A line of symmetry
2. (b) 2
3. (c) Isosceles Triangle
4. (d) F
5. (c) 6 (A regular hexagon has 6 sides)
6. (b) Circle
7. (a) 0
8. (d) Reflectional Symmetry
9. (c) D (Letters B, C, D, E, K have only horizontal lines of symmetry)
10. (b) Match exactly (Diagonals are lines of symmetry for a square)

Study these notes carefully. Understanding symmetry is about visualizing how shapes can be divided equally. Good luck with your preparation!